A circle has a radius of $8$. An arc in this circle has a central angle of $270^\circ$. What is the length of the arc? ${16\pi}$ ${270^\circ}$ $\color{#DF0030}{12\pi}$ ${8}$
Answer: First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (8) = 16\pi$ The ratio between the arc's central angle $\theta$ and $360^\circ$ is equal to the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{360^\circ} = \dfrac{s}{c}$ $\dfrac{270^\circ}{360^\circ} = \dfrac{s}{16\pi}$ $\dfrac{3}{4} = \dfrac{s}{16\pi}$ $\dfrac{3}{4} \times 16\pi = s$ $12\pi = s$